The Kerr-nonlinear parametric oscillator (KPO) provides a foundational semiclassical model for cat-state quantum hardware. Standard analyses of the KPO typically rely on autonomous, frozen-time approximations to describe the stabilization of macroscopic coherent states. However, state preparation and gate manipulation are driven by explicitly time-dependent microwave pulses, so the operational dynamics are inherently nonautonomous. In this paper, we show that static algebraic equilibrium pictures are incomplete for describing both state formation and gate-induced transport in the Kerr-cat qubit. For nonautonomous state preparation, we analyze the ramped resonant model by combining a linear nonautonomous stability analysis with a local invariant-graph reduction near the vacuum trajectory. This yields a quintic reduced normal form in the critical direction and identifies two symmetric post-threshold moving branches that organize the local state-formation dynamics. The associated diagnostics separate the reduced branch dynamics from the full two-dimensional phase-twist relaxation observed in the hardware coordinates. For gate execution, we model a fast pulse as a weak aperiodic perturbation of the conservative resonant figure-eight separatrix and apply Melnikov's method to derive a leading-order transport criterion. In this framework, transient lobe dynamics emerge as a semiclassical mechanism for non-adiabatic leakage, and the resulting amplitude-width threshold curve provides a leading-order geometric indicator for the onset of gate-pulse-induced transport.
The Kerr-nonlinear parametric oscillator (KPO) provides a foundational semiclassical model for cat-state quantum hardware. Standard analyses of the KPO typically rely on autonomous, frozen-time approximations to describe the stabilization of macroscopic coherent states. However, state preparation and gate manipulation are driven by explicitly time-dependent microwave pulses, so the operational dynamics are inherently nonautonomous. In this paper, we show that static algebraic equilibrium pictures are incomplete for describing both state formation and gate-induced transport in the Kerr-cat qubit. For nonautonomous state preparation, we analyze the ramped resonant model by combining a linear nonautonomous stability analysis with a local invariant-graph reduction near the vacuum trajectory. This yields a quintic reduced normal form in the critical direction and identifies two symmetric post-threshold moving branches that organize the local state-formation dynamics. The associated diagnostics separate the reduced branch dynamics from the full two-dimensional phase-twist relaxation observed in the hardware coordinates. For gate execution, we model a fast pulse as a weak aperiodic perturbation of the conservative resonant figure-eight separatrix and apply Melnikov's method to derive a leading-order transport criterion. In this framework, transient lobe dynamics emerge as a semiclassical mechanism for non-adiabatic leakage, and the resulting amplitude-width threshold curve provides a leading-order geometric indicator for the onset of gate-pulse-induced transport.
The Kerr-nonlinear Parametric Oscillator (KPO) provides a foundational semi-classical model for cat-state quantum hardware. Standard analyses of the KPO typically rely on autonomous, frozen-time approximations to describe the stabilization of macroscopic coherent states. However, state preparation and gate manipulation are driven by explicitly time-dependent microwave pulses, so the operational dynamics are inherently nonautonomous. In this paper, we show that static algebraic equilibrium pictures are incomplete for describing both state formation and gate-induced transport in the Kerr-cat qubit. For nonautonomous state preparation, we analyze the ramped resonant model by combining a linear nonautonomous stability analysis with a local invariant-graph reduction near the vacuum trajectory. This yields a quintic reduced normal form in the critical direction and identifies two symmetric post-threshold moving branches that organize the local state-formation dynamics. The associated diagnostics separate the reduced branch dynamics from the full two-dimensional phase-twist relaxation observed in the hardware coordinates. For gate execution, we model a fast pulse as a weak aperiodic perturbation of the conservative resonant figure-eight separatrix and apply Melnikov’s method to derive a leading-order transport criterion. In this framework, transient lobe dynamics emerge as a semi-classical mechanism for nonadiabatic leakage, and the resulting amplitude-width threshold curve provides a leading-order geometric indicator for the onset of gate-pulse-induced transport.
A Kerr-cat qubit encodes a logical bit in the two wells of a parametrically driven nonlinear oscillator, and a logic gate is a transient change of the drive. In the phase plane the gate deforms the double well and can split its separatrix into a turnstile that carries trajectories across the dividing surface between the wells; the same pulse, acting on the quantum oscillator, can corrupt the encoded bit. We study this process over a disk of coherent-state preparations, comparing classical phase-space transport diagnostics with the open-system quantum outcome on a common domain so that the two can be compared point by point. The central finding is that the corruption depends on the full temporal protocol, not on pulse strength alone: a sudden quench erases the bit, whereas a smooth ramp of the same peak amplitude largely preserves it. A finite-time sensitivity field locates the classical transport boundary, and a Loschmidt echo evaluated near the end of the gate predicts the much later quantum outcome. Sweeps of pulse amplitude and width, of cat size, and of engineered two-photon dissipation map where the classical transport picture predicts the quantum loss of the bit and where it does not.
A Kerr-cat qubit encodes a logical bit in the two wells of a parametrically driven nonlinear oscillator, and a logic gate is a transient change of the drive. In the phase plane, the gate deforms the double well and can split its separatrix into a turnstile that carries trajectories across the dividing surface between the wells; the same pulse, acting on the quantum oscillator, can corrupt the encoded bit. We study this process over a disk of coherent-state preparations, comparing classical phase-space transport diagnostics with the open-system quantum outcome on a common domain so that the two can be compared point by point. The central finding is that the corruption depends on the full temporal protocol, not on pulse strength alone: a sudden quench erases the bit, whereas a smooth ramp of the same peak amplitude largely preserves it. A finite-time sensitivity field locates the classical transport boundary, and a Loschmidt echo evaluated near the end of the gate predicts the much later quantum outcome. Sweeping the pulse amplitude, the pulse width, the cat size, and the rate of an engineered two-photon dissipation, we map where the classical transport picture predicts the quantum loss of the bit and where it does not.
A Kerr-nonlinear parametric oscillator (KPO) can generate a quantum superposition of two oscillating states, known as a Schr\"odinger cat state, via quantum adiabatic evolution and can be used as a qubit for gate-based quantum computing and quantum annealing. In this work, we investigate complex dynamics, i.e., chaos, in two coupled nondissipative KPOs at a few-photon level. After showing that a classical model for this system is nonintegrable and consequently exhibits chaotic behavior, we provide quantum counterparts for the classical results, which are quantum versions of the Poincar\'e surface of section and its lower-dimensional version defined with time integrals of the Wigner and Husimi functions and also the initial and long-term behavior of out-of-time-ordered correlators. We conclude that some of them can be regarded as quantum signatures of chaos, together with energy-level spacing statistics (conventional signature). Thus, the system of coupled KPOs is expected to offer not only an alternative approach to quantum computing but also a promising platform for the study on quantum chaos.
Kerr parametric oscillators (KPOs) can stabilize the superpositions of coherent states, which can be utilized as qubits, and are promising candidates for realizing hardware-efficient quantum computers. Although elementary gates for universal quantum computation with KPO qubits have been proposed, these gates are usually based on adiabatic operations and thus need long gate times, which result in errors caused by photon loss in KPOs realized by, e.g., superconducting circuits. In this work we accelerate the elementary gates by experimentally feasible control methods, which are based on numerical optimization of pulse shapes for shortcuts to adiabaticity. By numerical simulations, we show that the proposed methods can achieve speedups compared to adiabatic ones by up to six times with high gate fidelities of 99.9%. These methods are thus expected to be useful for quantum computers with KPOs.
Kerr parametric oscillators (KPOs), two-photon driven Kerr-nonlinear resonators, can stably hold coherent states with opposite-sign amplitudes and are promising devices for quantum computing. Recently, we have theoretically proposed a two-qubit gate Rzz for highly detuned KPOs and called it a conditional-driving gate [Chono et al., Phys. Rev. Res. 4, 043054 (2022)]. In this study, analyzing its superconducting-circuit model and deriving a corresponding static model, we find that an AC-Zeeman shift due to the flux pulse for the gate operation largely affects the gate performance. This effect becomes a more aggravating factor with shorter gate times, leading to an increase in the error rate. We thus propose a method to cancel this undesirable effect. Furthermore, through the use of shortcuts to adiabaticity and the optimization of flux pulses, we numerically demonstrate a conditional-driving gate with average fidelity exceeding 99.9% twice faster than that without the proposed cancellation method and the shortcuts to adiabaticity.